Run an external application with the bge
This task is fairly simple to do with a Python script.

You could add a python script as a controller to launch a browser:

import subprocess
subprocess.call(['pathToChrome\\chrome.exe',
                 'http://blender.stackexchange.com',
                 'Message'])




To save a screenshot you could use one of the command line capturing tools like:


webkit2png
command-line-program-to-create-website-screenshots-on-linux
you might add an os to google "browser screenshot command line"


Once you have the saved screenshot you could manipulate a plane to use the screenshot as a texture see createTexture in Blender API on how to dynamically change a textures image.